The theme is:
Mathematics.
First riddle: (correct answer)
An angle
Pills and pills, stuck between the hills.
Pills refers to the theta sign, because they look alike. The second part refers to the fact that angles are sandwhiched between two lines.
I may be right but must not be wrong.
Right angles exist but wrong angles do not.
I may be cute but will not be ferocious.
Acute angles exist but not ferocious angles.
I am from Greece. Oh! So precious.
The idea of angles originated from Euclidean geometry which is Greek. I am not sure about the second part though, I think this has something to do with the fact knowledge of angles is useful in construction and so is precious to humanity.
Second riddle:
The square root symbol.
Short pistols, long snipers,
It looks like a pistol with a smaller gun attached to it but with a greater barrel length to handle ratio (sniper).
From earth to heaven I form borders,
Perhaps something to do with this diagram: 
The optimists would be seperated,
The positive numbers (optimists) has two square roots, one negative and one positive.
The pessimists would call me an eye.
I don't like the way this clue is phrased but think this has to do with the fact that the square root of a negative number (pessimists) is an imaginary number which is denoted by the $i$ symbol hence the word 'eye'.
Third riddle:
The Exponent.
Flying on the sky, but I have no wings.
It floats on top of regular numbers in its notation with no visible 'wings' of course.
What I've made, undefined rings.
Wikipedia: $0^0$ is more complicated, and the choice of whether to assign it a value and what value to assign may depend on context.
Things will duplicate, swiftly,
Exponents makes numbers duplicate and then multiply with one another.
To huge strings I am calculated, roughly.
This hints at Power Series'.
Fourth riddle (correct answer)
This is the the factorial sign !
I yell, to everyone,
It is an exclamation mark
I will give you something, even you have none.
0! = 1
From series I appear, neither Asian nor European, thus forming, the Pie Game
$$\pi=\sum_{n=0}^{\infty}{(25n-3)n!(2n)!\over2^{n-1}(3n)!}$$ is due to the American mathematician, Bill Gosper. This answer also implies an interpretation of the word pie which alludes to the phrase American Pie hence a pie game, which means the equation has a 'play' on the word pie if you look at it this way (pie has two ways of being significant to this equation).
